Deep learning accelerated solutions of incompressible Navier-Stokes equations on non-uniform Cartesian grids

TL;DR

This paper introduces an advanced deep learning framework using Mesh-Conv operators to efficiently solve the pressure Poisson equation in incompressible flow simulations on non-uniform Cartesian grids, enhancing convergence and generalization.

Contribution

It extends the HyDEA method to non-uniform grids by integrating Mesh-Conv operators and a multi-level grid spacing strategy, enabling accurate and robust flow simulations near complex geometries.

Findings

MConv-based HyDEA outperforms traditional iterative solvers in convergence speed.

The method generalizes well across different obstacle geometries without retraining.

Benchmark tests show significant improvements on non-uniform grids.

Abstract

The pressure Poisson equation (PPE) represents the primary computational bottleneck in fractional step methods for incompressible flow simulations, requiring iterative solutions of large-scale linear systems. We previously introduced HyDEA, a hybrid approach to accelerate the PPE solution process. However, its current implementation is limited to uniform Cartesian grids. Accurately resolving complex flow dynamics near solid boundaries requires local grid refinement, yet extending the original HyDEA to non-uniform Cartesian grids is fundamentally challenging, as its standard convolution operators are inherently ill-suited for processing data with spatially varying resolutions. To address this limitation, we adopt the Mesh-Conv (MConv) operator, which explicitly incorporates grid spacing information into convolution operations. Specifically, MConv operator replaces a subset of the standard convolution operators within the U-Net-based branch network of the deep operator network, with the necessary grid spacing information computed via a novel multi-level distance vector map construction strategy. Building upon this enhanced architecture, the framework seamlessly extends to simulate flows interacting with solid structures using a decoupled immersed boundary projection method. Furthermore, by training exclusively on fabricated linear systems rather than conventional flow-dependent datasets, the model generalizes effortlessly across diverse immersed obstacle geometries with fixed neural network weights. Benchmark results demonstrate that the MConv-based HyDEA significantly outperforms both standalone preconditioned conjugate gradient methods and the standard convolution-based HyDEA in convergence performance on strongly non-uniform Cartesian grids. The robustness and generalizability of the MConv-based HyDEA underscore its potential for real-world computational fluid dynamics applications.